A Teacher Used The Change Of Base Formula To Determine Whether The Equation Below Is Correct:$\left(\log _2 10\right)\left(\log _4 8\right)\left(\log _{10} 4\right) = 3$Which Statement Explains Whether The Equation Is Correct?A. The Equation

Introduction

In mathematics, the change of base formula is a fundamental concept used to simplify logarithmic expressions. It allows us to express a logarithm in terms of another base, making it easier to work with and solve equations. In this article, we will explore how a teacher used the change of base formula to determine whether a given equation is correct. We will delve into the details of the equation, apply the change of base formula, and provide a clear explanation of the solution.

The Equation

The equation in question is:

$$\left(\log _2 10\right)\left(\log _4 8\right)\left(\log _{10} 4\right) = 3$$

This equation involves logarithms with different bases, and the teacher wants to determine whether it is correct. To do this, we need to apply the change of base formula to simplify the expression.

Applying the Change of Base Formula

The change of base formula states that:

$$\log _b a = \frac{\log _c a}{\log _c b}$$

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$. We can use this formula to rewrite each logarithm in the equation in terms of a common base.

Let's start with the first logarithm:

$$\log _2 10 = \frac{\log _c 10}{\log _c 2}$$

We can choose any base $c$ we like, but let's choose base 10 for simplicity. Then:

$$\log _2 10 = \frac{\log _{10} 10}{\log _{10} 2} = \frac{1}{\log _{10} 2}$$

Similarly, for the second logarithm:

$$\log _4 8 = \frac{\log _{10} 8}{\log _{10} 4} = \frac{\log _{10} 2^3}{\log _{10} 2^2} = \frac{3}{2}$$

And for the third logarithm:

$$\log _{10} 4 = \frac{\log _{10} 2^2}{\log _{10} 2} = 2$$

Now we can substitute these expressions back into the original equation:

$$\left(\frac{1}{\log _{10} 2}\right)\left(\frac{3}{2}\right)(2) = 3$$

Simplifying the Equation

Let's simplify the equation by canceling out the common factors:

$$\frac{1}{\log _{10} 2} \cdot 3 \cdot 2 = 3$$

$$\frac{6}{\log _{10} 2} = 3$$

Now we can solve for $\log _{10} 2$:

$$\log _{10} 2 = \frac{6}{3} = 2$$

Conclusion

The equation $\left(\log _2 10\right)\left(\log _4 8\right)\left(\log _{10} 4\right) = 3$ is not correct. The correct value of $\log _{10} 2$ is 2, not 3.

Why the Equation is Incorrect

The equation is incorrect because the change of base formula was applied incorrectly. The change of base formula states that $\log _b a = \frac{\log _c a}{\log _c b}$, but in this case, the equation was simplified incorrectly. The correct simplification would have resulted in a value of 2 for $\log _{10} 2$, not 3.

Real-World Applications

The change of base formula has many real-world applications in fields such as engineering, physics, and computer science. It is used to simplify complex logarithmic expressions and solve equations involving different bases.

Conclusion

In conclusion, the teacher used the change of base formula to determine whether the equation $\left(\log _2 10\right)\left(\log _4 8\right)\left(\log _{10} 4\right) = 3$ is correct. By applying the change of base formula and simplifying the equation, we found that the equation is incorrect. The correct value of $\log _{10} 2$ is 2, not 3. This example illustrates the importance of applying mathematical formulas correctly and the value of using the change of base formula to simplify complex logarithmic expressions.

Final Thoughts

Introduction

In our previous article, we explored how a teacher used the change of base formula to determine whether a given equation is correct. We applied the change of base formula, simplified the equation, and found that the equation is incorrect. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the change of base formula.

Q&A

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to express a logarithm in terms of another base. It is given by:

$$\log _b a = \frac{\log _c a}{\log _c b}$$

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.

Q: Why do we need the change of base formula?

A: We need the change of base formula to simplify complex logarithmic expressions and solve equations involving different bases. It allows us to express a logarithm in terms of a common base, making it easier to work with and solve equations.

Q: How do we apply the change of base formula?

A: To apply the change of base formula, we need to:

1. Choose a common base $c$.
2. Rewrite each logarithm in the equation in terms of the common base $c$.
3. Simplify the equation using the change of base formula.

Q: What are some real-world applications of the change of base formula?

A: The change of base formula has many real-world applications in fields such as:

• Engineering: to simplify complex logarithmic expressions and solve equations involving different bases.
• Physics: to calculate the logarithm of a quantity in terms of a common base.
• Computer Science: to simplify complex logarithmic expressions and solve equations involving different bases.

Q: What are some common mistakes to avoid when applying the change of base formula?

A: Some common mistakes to avoid when applying the change of base formula include:

• Choosing an incorrect common base.
• Simplifying the equation incorrectly.
• Failing to check the validity of the equation.

Q: How can I practice using the change of base formula?

A: You can practice using the change of base formula by:

• Solving logarithmic equations involving different bases.
• Simplifying complex logarithmic expressions.
• Applying the change of base formula to real-world problems.

Conclusion

In conclusion, the change of base formula is a powerful tool for simplifying logarithmic expressions and solving equations involving different bases. By understanding and applying this formula correctly, we can solve complex mathematical problems and make accurate predictions in various fields. We hope this Q&A section has helped clarify any doubts and provided additional insights into the change of base formula.

Final Thoughts

The change of base formula is a fundamental concept in mathematics that has many real-world applications. By mastering this formula, we can simplify complex logarithmic expressions and solve equations involving different bases. We encourage you to practice using the change of base formula and explore its many applications in various fields.